Switching Apple TV on and off while not being at home

I have the latest generation Apple TV, and it’s incorrectly connected to HomeKit. I would like to be able to turn the TV on and off remotely when I’m not on the Wi-Fi network, accessing it only via 4G or 5G. I use the TV to create the impression that I’m at home. (My house is very visible from the road, and I want to give the impression that someone is home during my absence). I specifically want to reliably turn my Apple TV on and off when I’m not on the Wi-Fi network. I haven’t managed to get this to work yet, and I’ve been trying for a year—thankfully, not continuously. It seems like something that should be possible, but I simply can’t get it to work. I’m using an Apple iPhone 15 Pro, which is also properly configured with HomeKit.

I have serveral AppleTV's and one is configured to act as The Hub.

If I go to HomeKit on my iPhone 15 Pro, and click on that specific Apple TV, the first weird thing is that the Podcast show up as only option. If I'm in the WiFi cloud, and click on 'play' the podcast is indeed played at the Apple TV and hence at the connected TV. And I can switch off the AppleTV and in doing so, switch off the connected TV. So far, so good!

But If I'm outside of my house and would like to do the same, there is no way to get the AppleTV to obey me. Personally, I thought the whole idea of HomeKit was to be able to manage your Home while you where out of your home. The WiFi is done in my house with a bunch of Airport Extremes, no clue if that's relevant.

If it's not possible, fine, then I thank the community for saving me tons of time (although I've already spent way too much trying to get this to work. But I feel it should be possible, right?

Cheers,

PaulH

(It saddens me to say that the only things I can access while not at home are my Android-prepared Sony TV and my Philips Hue lights. But Apple....no luck).